Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 219
We prove now the estimate ν{|ψ ′′ | > b} ≤ce −δ b 0 . It is enough to consider the
case where b = 2l for some positive integer l. Recall that for simplicity we assumed
|ψ|≤1. It follows that |E(ψ|Fn)|≤1 and hence |Λ n g (ψ)|≤1. We have
|ψ ′′ |≤∑ |Λ
n≥1
n −2l
g (ψ)|≤δ1 ∑ δ
n≥1
2n
1 |Λ n g (ψ)| + ∑ |Λ
1≤n≤l
n g
Consequently,
ν � |ψ ′′ | > 2l � ≤ ν � ϕ > δ 2l�
−αδ
1 � e 2l
1 .
−2l
(ψ)|≤δ1 ϕ + l.
It is enough to choose δ0 < δ1 and c large enough. ⊓⊔
Lemma 1.99. The coboundary ψ ′′ − ψ ′′ ◦ g satisfies the LDT.
Proof. Given a function φ ∈ L 1 (μ), recall that Birkhoff’s sum SN(φ) is defined by
N−1
S0(φ) := 0 and SN(φ) :=
∑
n=0
φ ◦ g n
for N ≥ 1.
Observe that SN(ψ ′′ − ψ ′′ ◦ g) =ψ ′′ − ψ ′′ ◦ gN . Consequently, for a given ε > 0,
using the invariance of ν, wehave
ν � |SN(ψ ′′ − ψ ′′ ◦ g)| > Nε � �
≤ ν |ψ ′′ ◦ g N | > Nε
� �
+ μ |ψ
2
′′ | > Nε
�
2
�
= 2ν |ψ ′′ | > Nε
�
.
2
Lemma 1.98 implies that the last expression is smaller than e −Nhε for some hε > 0
and for N large enough. This completes the proof. ⊓⊔
It remains to show that ψ ′ satisfies the weak LDT. We use the following lemma.
Lemma 1.100. For every b ≥ 1, there are Borel sets WN such that ν(WN) ≤ cNe−δ b 0
and
�
e λ SN(ψ ′ �
) e
dν ≤ 2
−λ b �N + eλ b
,
2
X\WN
where c > 0 is a constant independent of b.
Proof. For N = 1, define W := {|ψ ′ | > b}, W ′ := g(W) and W1 := g −1 (W ′ ). Recall
that the Jacobian of ν is bounded by some constant κ. This and Lemma 1.98 imply
that
ν(W1)=ν(W ′ )=ν(g(W)) ≤ κν(W) ≤ ce −δ b 0
for some constant c > 0. We also have
�
e
X\W1
λ S1(ψ ′ �
)
dν = e
X\W1
λψ′
dν ≤ e λ b �
≤ 2
So, the lemma holds for N = 1.
e−λ b λ b + e
2
�
.